A generalization of second variation (Q1078688)

For \(1\leq p<\infty\), let \[ V^{(2)}_{2,p}(f;a,b)=\sup (\sum^{n- 1}_{i=0}| \frac{f(x_{2i+3})-f\quad (x_{2i+2})}{x_{2i+3}- x_{2i+2}}-\frac{f(x_{2i+1})-f(x_{2i})}{x_{2i+1}-x_{2i}}|^ p)^{1/p}, \] where the supremum is taken over all even subdivisions \((x_ i)_ 0^{2n+1}\) of [a,b]. For \(p=\infty\), let \[ V^{(2)}_{2,\infty}(f;a,b)=\sup_{a\leq x_ 0<x_ 1<x_ 2<x_ 3\leq b}| \frac{f(x\quad_ 3)-f(x_ 2)}{x_ 3-x_ 2}-\frac{f(x_ 1)- f(x_ 0)}{x_ 1-x_ 0}|. \] The main result: Let \(1\leq p\leq \infty\) and \(V^{(2)}_{2,p}(F;a,b)<\infty.\) Then F' is almost equal to a function f, with \(V_{1,p}(f;a,b)<\infty,\) such that \(V^{(2)}_{2,p}(F;a,b)=V_{1,p}(f;a,b).\) It seems that the above alternative definition is the natural generalization of Riesz' second variation [\textit{F. Riesz}, Ann. Sci. École Norm. Supér., III. Sér. 28, 33-68 (1911)]. For details we have to refer to the well organized and clearly written paper.